#### Answer

$x=\left\{ \dfrac{7}{3},3 \right\}$

#### Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|-3x+8|=1
,$ is equivalent to
\begin{array}{l}\require{cancel}
-3x+8=1 \text{ OR } -3x+8=-1
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
-3x+8=1
\\\\
-3x=1-8
\\\\
-3x=-7
\\\\
x=\dfrac{-7}{-3}
\\\\
x=\dfrac{7}{3}
\\\\\text{ OR }\\\\
-3x+8=-1
\\\\
-3x=-1-8
\\\\
-3x=-9
\\\\
x=\dfrac{-9}{-3}
\\\\
x=3
.\end{array}
Hence, the solutions are $
x=\left\{ \dfrac{7}{3},3 \right\}
.$